p-torsion of fields

In this form we can extend the Frobenius and hence this notion of pp-torsion from abelian groups to fields if we require our field to be of characteristic pp such that we have (a+b)n=an+bn(a+b)^n=a^n+b^n.

p-divisible groups

Sometimes the information encoded in the colimit Tp(G)=colimnG[pn]T_p(G)=colim_n G[p^n] (we passed a contravariant functor from rings to schemes) is considered to be not sufficient and one wants more generally to study the codirected system

We have as cardinality (in group theory also called “rank”) of the first item of the sequence cardkerp=phcard \ker \,p=p^h for some natural number hh. By pars pro toto we call php^h also the rank of the whole sequence and hh we call its height.

Conversely we can define a pp-divisible group to be a codirected diagram